A055022 Number of 1-punctured staircase polygons (by perimeter) with a hole of perimeter 4.

0, 0, 0, 0, 0, 0, 0, 0, 1, 12, 92, 576, 3214, 16664, 82160, 390656, 1807781, 8192524, 36519556, 160645504, 699030226, 3014470024, 12901501696, 54863119744, 232022899306, 976598630968, 4093581923320, 17096805375360, 71176501409756

Offset

0,10

References

Christoph Richard, Iwan Jensen, and Anthony J. Guttmann, Area distribution and scaling function for punctured polygons, Electronic journal of combinatorics 15 (2008), #R53. See Equation (23).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..500

A. J. Guttmann et al., Punctured polygons and polyominoes on the square lattice, arXiv:cond-mat/0003441 [cond-mat.stat-mech], 2000.

A. J. Guttmann et al., Punctured polygons and polyominoes on the square lattice, J. Physics A: Math. and Gen, 33 (9) (2000), 1735-1764.

Formula

G.f.: (2*x^4 - 16*x^3 + 20*x^2 - 8*x + 1)/(2*(1 - 4*x)) - (1 - 6*x + 10*x^2 - 4*x^3)/(2*sqrt(1 - 4*x)). - N. J. A. Sloane, Sep 10 2024

D-finite with recurrence n*(n-8)*a(n) +2*(-4*n^2+35*n-45)*a(n-1) +8*(2*n-9)*(n-5)*a(n-2)=0. - R. J. Mathar, Aug 14 2012

For n>3, a(n) = 4^(n-4)-binomial(2n,n)(n-3)(n^2-5n+10)/(4(2n-1)(2n-3)(2n-5)). - Michael D. Weiner, Jan 17 2018

Maple

gf := (2*x^4 - 16*x^3 + 20*x^2 - 8*x + 1)/(2*(1 - 4*x)) - (1 - 6*x + 10*x^2 - 4*x^3)/(2*sqrt(1 - 4*x)):
s := series(gf, x, 50):
for i from 0 to 50 do printf(`%d, `, coeff(s, x, i)) od:

Mathematica

Join[{0, 0, 0, 0}, Table[4^(n - 4) - Binomial[2 n, n] (n - 3) (n^2 - 5 n + 10) / (4 (2 n - 1) (2 n - 3) (2 n - 5)), {n, 4, 50}]] (* Vincenzo Librandi, Jan 20 2018 *)

Prog

(Magma) [0, 0, 0, 0] cat [4^(n-4)-Binomial(2*n, n)*(n-3)*(n^2-5*n+10) div (4*(2*n-1)*(2*n-3)*(2*n-5)): n in [4..30]]; // Vincenzo Librandi, Jan 20 2018

Crossrefs

Cf. A055024 (hole perimeter 6).

Sequence in context: A246585 A120990 A220330 * A220937 A221291 A044263

Adjacent sequences: A055019 A055020 A055021 * A055023 A055024 A055025

Keyword

easy,nonn

Author

James A. Sellers, May 31 2000

Status

approved